Switzerland were on the verge of making it two wins out of two at the start of their Euro 2020 qualifying campaign but Dalsgaard outjumped goalkeeper Yann Sommer to head home and earn Denmark an astonishing and unlikely point.
The home team broke the deadlock in the 19th minute when Remo Freuler beat Denmark keeper Kasper Schmeichel with a cool finish for his first international goal.
Swiss captain Granit Xhaka hammered home a 20-metre shot in the 66th minute and Breel Embolo made it 3-0 from close range after Manuel Akanji jumped highest to head across goal.
Yet Denmark came back from the dead in the dying moments of the match with goals by Mathias Jorgensen, substitute Christian Gytkjaer and Dalsgaard in less than 10 minutes.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
In the other game in Group D, Ireland produced a dominant display to beat Georgia 1-0 after Conor Hourihane gave keeper Giorgi Loria no chance with a brilliant free kick in Mick McCarthy's first home game since he returned for his second spell as Ireland coach.
Ireland, who battled to a 1-0 win away to rank outsiders Gibraltar in blustery conditions in their opening match on Saturday, lead the way with six points from two games. Switzerland, who beat Georgia 2-0 in Tbilisi in their opener, are second with four points.
Switzerland visit Ireland on Sept. 5 in their next match in the group while Ireland are away to Denmark on June 7.
(Reporting by Angel Krasimirov; Editing by Toby Davis)
